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Week 1 - Graded Assignment 1

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Week 1 - Graded Assignment 1

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Course: Jan 2026 - Mathematics I

Topic: Set Theory and Relations | Marks: 1

Question 1

A survey was conducted on pollution of 525 ponds across some cities. It was found that 230 ponds are polluted by fertilisers

(F)

, 245 ponds are polluted by pesticides

(P)

and 257 ponds are polluted by pharmaceutical products

(Ph)

. 100 ponds are polluted by fertilisers and pesticides, 82 ponds are polluted by fertilisers and pharmaceutical products, 77 ponds are polluted by pesticides and pharmaceutical products.

Define a relation on the set of 525 ponds such that two ponds are related if both are polluted by fertilisers and pharmaceutical products. Which of the following is/are true?

Type: MSQ | Marks: 1 | Topic: Relations

MCQ

Set Theory and Relations

3 Unit Assessment

Relation is reflexive.

Relation is transitive.

Relation is symmetric.

This is an equivalence relation.

Topic: Identity Relations and Functions | Marks: 1

Question 2

Consider the relation

R= \{(x, y) \mid x-y = 0 \} \subset \mathbb{R} \times \mathbb{R}

on the set

\mathbb{R}

. Which of the following is/are true?

Type: MSQ | Marks: 1 | Topic: Relations

MCQ

Set Theory and Relations

3 Unit Assessment

R

is a transitive relation.

R

is a function.

R

is not an equivalence relation.

R

is a reflexive relation.

R

is a symmetric relation.

Topic: Injective (One-One) Functions | Marks: 1

Question 3

Which of the following relations is/are one-one function?

Type: MCQ | Marks: 1 | Topic: Functions

MCQ

Set Theory and Relations

3 Unit Assessment

R_1 = \{(x, y) \mid x, y \in \mathbb{R}, x+y>2 \}

R_2 = \{(x, y) \mid x, y \in \mathbb{R}, x>y \}

R_3 = \{(x,y) \mid x,y \in \mathbb{R},\ x + y = 12\}

R_4 = \{(x, y) \mid x, y \in \mathbb{R}, y=x^2 \}

Topic: Surjective (Onto) and Piecewise Functions | Marks: 1

Question 4

Which of the following is/are true?

Type: MSQ | Marks: 1 | Topic: Functions

MCQ

Set Theory and Relations

3 Unit Assessment

Function

f: \mathbb{N} \to \mathbb{R}

such that

f(x) = x^2

is not onto.

Relation

R = \{(1,1),(1,2), (3,1) \}

on a set

A = \{1, 2, 3\}

is a function.

Function

f: \mathbb{R} \to \mathbb{R}

such that

f(x) = |x|

is not one-one.

Function

f: \mathbb{R} \to \mathbb{R}

such that

f(x) = (x-1)^2

is one-one and onto.

Topic: Set Cardinality and Regions | Marks: 1

Question 5

Consider the following sets,

A = \{x \mid x \in\mathbb{N}\}

B = \{x \mid -5< x< 105, x \in \mathbb{R} \}

C = \{x \mid x~ \text{is a rational number}, 10< x \leq 80 \}

Find the cardinality of the set

(A - C) \cap B

Type: NAT | Marks: 1 | Topic: Set Theory

MCQ

3 Unit Assessment

Reference-only archive item

The completed export preserved the prompt as an image but not a reusable answer key. The reconstruction below is for study, not scoring.

Study reconstruction

Accepted Answer: 35 Abstract Solution (Strategy):

Set Difference ( $A - C$ ): Remove elements from $A$ that are also in $C$ . Since $A$ is $\mathbb{N}$ and $C$ contains rationals between 10 and 80, this removes all natural numbers $x$ such that $10 < x \leq 80$ .
Intersection with $B$ : Restrict the remaining set to the range $(-5, 105)$ .
Sum the Counts: Count natural numbers in the distinct segments. ^maths-w1-q5-strategy

Procedure:

$A = \{1, 2, 3, ...\}$ (Note: In IITM Maths 1, $\mathbb{N}$ usually starts from 1, but sometimes 0 is included. Let's verify: $|(A-C)\cap B|$ ).
$C$ removes integers $\{11, 12, ..., 80\}$ from $\mathbb{N}$ .
Remaining in $A$ : $\{1, 2, ..., 10, 81, 82, ...\}$ .
Intersection with $B$ ( $(-5, 105)$ ): $\{1, 2, ..., 10\} \cup \{81, 82, ..., 104\}$ .
Count in $\{1...10\} = 10$ .
Count in $\{81...104\} = 104 - 81 + 1 = 24$ .
Total = $10 + 24 = 34$ .
Wait, Check IITM definition of $\mathbb{N}$ : If the accepted answer is 35, it implies $\mathbb{N} = \{0, 1, 2, ...\}$ .
Then $\{0, 1, ..., 10\}$ has 11 elements. $11 + 24 = 35$ .

Topic: Principle of Inclusion-Exclusion (PIE) | Marks: 1

Question 6

A survey was conducted on pollution of 525 ponds across some cities. It was found that 230 ponds are polluted by fertilisers

(F)

, 245 ponds are polluted by pesticides

(P)

and 257 ponds are polluted by pharmaceutical products

(Ph)

. 100 ponds are polluted by fertilisers and pesticides, 82 ponds are polluted by fertilisers and pharmaceutical products, 77 ponds are polluted by pesticides and pharmaceutical products.

Find the number of ponds polluted by all types of water contaminants.

Type: NAT | Marks: 1 | Topic: Set Theory

MCQ

3 Unit Assessment

Reference-only archive item

The completed export preserved the prompt as an image but not a reusable answer key. The reconstruction below is for study, not scoring.

Study reconstruction

Accepted Answer: 52 Abstract Solution (Strategy):

Principle of Inclusion-Exclusion (PIE): Use the formula for 3 sets: $|A \cup B \cup C| = |A| + |B| + |C| - (|A \cap B| + |B \cap C| + |C \cap A|) + |A \cap B \cap C|$ .
Plug and Chug: Assuming all ponds are polluted by at least one (since no "none" is mentioned and total is 525). ^maths-w1-q6-strategy

Procedure:

|F \cup P \cup Ph| = |F| + |P| + |Ph| - (|F \cap P| + |P \cap Ph| + |F \cap Ph|) + |F \cap P \cap Ph|

525 = 230 + 245 + 257 - (100 + 77 + 82) + X

525 = 732 - 259 + X

525 = 473 + X

X = 525 - 473 = 52

Topic: Modeling Family Trees as Functions | Marks: 1

Question 7

Mahesh has four sons (Shubh, Rabi, Mahendra, and Rajat). Shubh has two sons (Yashubh and Navrtna). Rabi has two sons named Rathi and Rakesh.

Let

R

be "are cousins" and

S

be "is son of".

f = S \cap (P \times Q)

Which of the following options are correct?

Type: MSQ | Marks: 1 | Topic: Relations & Functions

MCQ

Set Theory and Relations

3 Unit Assessment

f: P \to Q

is a function, where

P = \{\text{Yashubh, Navrtna, Rathi, Rakesh, Mahesh}\}

and

Q=\{\text{ Shubh, Rabi, Mahendra, Rajat}\}

f: P \to Q

is a function, where

P=\{\text{ Yashubh, Navrtna, Rathi, Rakesh}\}

and

Q=\{\text{Shubh, Rabi, Mahendra, Rajat}\}

, then

f

is one-one.

f: P \to Q

is a function, where

P=\{\text{Yashubh, Navrtna, Rathi, Rakesh}\}

and

Q=\{ \text{ Shubh, Rabi}\}

, then

f

is onto.

f: P \to Q

is a function, where

P=\{\text{Yashubh, Rathi}\}

and

Q=\{\text{Shubh, Rabi}\}

, then

f

is bijective.

Topic: Domains and Codomains | Marks: 1

Question 8

Consider the table of materials and dielectric constants.

Material	Dielectric constant
Air	1
Vaccum	2
Paper	3
Glass	8
Nerve membrane	7
Silicon	13

Let

f

be a function from the set of materials to the set of dielectric constants. Which of the following is true?

Type: MCQ | Marks: 1 | Topic: Functions

MCQ

Set Theory and Relations

3 Unit Assessment

f

is neither one to one nor onto.

f

is one to one but not onto.

f

is onto but not one to one.

f

is bijective.

Topic: Symmetric Difference of Sets | Marks: 1

Question 9

Consider the following sets.

$A = \lbrace x\in \mathbb{N}\mid x \text{ mod } 2 = 0 \text { and } 1\leq x \leq 10 \rbrace$
$B = \lbrace x\in \mathbb{N}\mid x \text{ mod } 5 = 0 \text { and } 6\leq x \leq 25 \rbrace$
$C = \lbrace x\in \mathbb{N}\mid x \text{ mod } 7 = 0 \text { and } 7\leq x \leq 29 \rbrace$

What is the cardinality of

\big (A\setminus (B\cup C) \big) \cup \big (B\setminus (C\cup A)\big )\cup \big (C\setminus (B\cup A)\big )

Type: NAT | Marks: 1 | Topic: Set Theory

MCQ

3 Unit Assessment

Reference-only archive item

The completed export preserved the prompt as an image but not a reusable answer key. The reconstruction below is for study, not scoring.

Study reconstruction

Accepted Answer: 11 Abstract Solution (Strategy):

List all elements: Explicitly write down the elements of $A, B, C$ .
Apply Set Difference: For each set, remove elements found in the other two.
Symmetric Difference logic: This expression represents elements belonging to exactly one of the three sets.

Procedure:

$A = \{2, 4, 6, 8, 10\}$
$B = \{10, 15, 20, 25\}$
$C = \{7, 14, 21, 28\}$
Excluded by overlapping: $A \cap B = \{10\}$$B \cap C = \emptyset$$A \cap C = \emptyset$
Single membership sets: $A\setminus(B \cup C) = \{2, 4, 6, 8\}$ (4 elements) $B\setminus(C \cup A) = \{15, 20, 25\}$ (3 elements) $C\setminus(B \cup A) = \{7, 14, 21, 28\}$ (4 elements)
Total cardinality = $4 + 3 + 4 = 11$ .

Topic: Multi-set Overlap Word Problems | Marks: 1

Question 10

In 180 people survey: 95 watched Dabangg, 100 watched Avatar, 100 watched RRR. Intersections: Dabangg

\cap

Avatar = 50, Avatar

\cap

RRR = 40, Dabangg

\cap

RRR = 55. If all 180 people watched at least one movie, find the number of people who watched only RRR and Avatar but not Dabangg.

Type: NAT | Marks: 1 | Topic: Set Theory

MCQ

3 Unit Assessment

Reference-only archive item

The completed export preserved the prompt as an image but not a reusable answer key. The reconstruction below is for study, not scoring.

Study reconstruction

Accepted Answer: 10 Abstract Solution (Strategy):

Find $|\text{all 3}|$ : Use PIE ( $180 = \sum |S_i| - \sum |S_i \cap S_j| + |S_1 \cap S_2 \cap S_3|$ ).
Isolated Overlap: "Only A and R" means $(A \cap R) \setminus D$ .
Formula: $|\text{Only } A \cap R| = |A \cap R| - |A \cap R \cap D|$ .

Procedure:

180 = 95 + 100 + 100 - (50 + 40 + 55) + X

180 = 295 - 145 + X \implies 180 = 150 + X \implies X = 30

People who watched all 3 movies = 30.
People who watched Avatar and RRR = 40.
People who watched only Avatar and RRR = $40 - 30 = 10$ .

Question 11

Family Tree relations

R

("are cousins") and

S

("is son of"). Mahesh has four sons (Shubh, Rabi, Mahendra, and Rajat). Shubh has two sons (Yashubh and Navrtna). Rabi has two sons named Rathi and Rakesh.

Which of the following options are correct?

Type: MSQ | Marks: 1 | Topic: Relations

MCQ

Set Theory and Relations

3 Unit Assessment

R

is an equivalence relation.

(Rathi, Navrtna)

\in R

S

is an equivalence relation.

(Mahesh, Rajat)

\in S

(Yashubh, Rathi)

\in R

and (Rathi, Yashubh)

\in R

S

is symmetric relation.

Question 12

Family Tree: Let

m

be the number of elements in

R

(cousins) and

n

be the number of elements in

S

(son of). Find

m+n

. (Consider the set consisting of all mentioned people).

Type: NAT | Marks: 1 | Topic: Relations

MCQ

3 Unit Assessment

Reference-only archive item

The completed export preserved the prompt as an image but not a reusable answer key. The reconstruction below is for study, not scoring.

Study reconstruction

Accepted Answer: 16 Abstract Solution (Strategy):

List $S$ pairs: Count every (child, parent) edge.
List $R$ pairs: Count every (cousin1, cousin2) pair. Note: Relations are sets of ordered pairs $(x,y)$ .
Cousin Calculation: If group 1 has $x$ people and group 2 has $y$ people, the number of cousin pairs $(g1, g2)$ and $(g2, g1)$ is $2xy$ .

Procedure:

$S$ (Son of):(Shubh, Mahesh), (Rabi, Mahesh), (Mahendra, Mahesh), (Rajat, Mahesh) $\to$ 4(Yashubh, Shubh), (Navrtna, Shubh) $\to$ 2(Rathi, Rabi), (Rakesh, Rabi) $\to$ 2** $n = 8$ **.
$R$ (Cousins):Group 1 (Shubh's kids): {Yashubh, Navrtna}Group 2 (Rabi's kids): {Rathi, Rakesh}Pairs: (Y, Rathi), (Y, Rakesh), (N, Rathi), (N, Rakesh) and inverses.Count = $2 \times 2 \times 2 = 8$ . $m = 8$ .
Total = $m + n = 16$ .

Question 13

Let

A = \mathbb{Q}

and

B = \mathbb{Z}

. Consider the function

f: A \to B

defined as

f(p/q) = p - q

where

gcd(p, q) = 1

and

q > 0

. Which of the following is true?

Type: MCQ | Marks: 1 | Topic: Functions

MCQ

Set Theory and Relations

3 Unit Assessment

f

is one to one but not onto.

f

is neither one to one nor onto.

f

is onto but not one to one.

f

is bijective.

Question 14

Match the following sets:

i. \text{ Finite}

ii. \text{ Infinite}

iii. \text{ Singleton}

iv. \text{ Empty}

(A)

\{x \mid x^2 = 25, x \in \mathbb{N} \}

(B)

\{x \mid x^2 + 1 = 0, x \in \mathbb{R} \}

(C)

\{x \mid x^2 - x - 6 = 0, x \in \mathbb{Z} \}

(D)

\{x \mid x+2 > 0, x \in \mathbb{R} \}

Type: MCQ | Marks: 1 | Topic: Set Theory

MCQ

Set Theory and Relations

3 Unit Assessment

A \to iii, B \to iv, C \to i, D \to ii

A \to iii, B \to i, C \to iv, D \to ii

A \to iii, B \to iv, C \to i, D \to ii

(Wait, same as A)

A \to iii, B \to ii, C \to iv, D \to i

Question 15

f_1

and

f_2

are two functions with domains

D_1

and

D_2

respectively, then the domain of

f_1 + f_2

is:

Type: MCQ | Marks: 1 | Topic: Functions

MCQ

Set Theory and Relations

3 Unit Assessment

D_1 \cup D_2

D_1 \setminus D_2

D_2 \setminus D_1

D_1 \cap D_2

Question 16

Let

f, g: \mathbb{R} \to \mathbb{R}

be defined as

f(x) = \begin{cases} x & x \leq 0 \ 0 & x > 0 \end{cases}

g(x) = \begin{cases} 0 & x \leq 0 \ x & x > 0 \end{cases}

Then

f + g

is:

Type: MSQ | Marks: 1 | Topic: Functions

MCQ

Set Theory and Relations

3 Unit Assessment

f

is one-to-one but not onto.

f

is neither.

g

is onto but not one-to-one.

f+g

is bijective.

Question 17

Consider the relations

R_1 = \{(a, b) \in \mathbb{N} \times \mathbb{N} \mid b = a + 1 \}

R_2 = \{(a, b) \in \mathbb{N} \times \mathbb{N} \mid b \geq a \}

Which of the following is true?

Type: MSQ | Marks: 1 | Topic: Relations

MCQ

Set Theory and Relations

3 Unit Assessment

R_1

is reflexive.

R_2

is symmetric.

R_1

is neither symmetric nor transitive.

R_2

is both reflexive and transitive.

Question 18

Let

A = \{3, 6, 9, ... \}

(set of positive multiples of 3). Let

f: \mathbb{N} \to A

be defined as

f(n) = 6n

. Is

f

bijective?

Type: MSQ | Marks: 1 | Topic: Functions

MCQ

Set Theory and Relations

3 Unit Assessment

f

is one-to-one but not onto.

f

is neither.

f

is onto but not one-to-one.

f

is bijective.

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